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September  2021, 29(4): 2553-2560. doi: 10.3934/era.2021001

Note on coisotropic Floer homology and leafwise fixed points

Fabian Ziltener

Utrecht University, Mathematics Institute, Budapestlaan 6, 3584 CD Utrecht, The Netherlands

Received  November 2019 Revised  September 2020 Published  September 2021 Early access  January 2021

For an adiscal or monotone regular coisotropic submanifold $ N $ of a symplectic manifold I define its Floer homology to be the Floer homology of a certain Lagrangian embedding of $ N $. Given a Hamiltonian isotopy $ \varphi = ( \varphi^t) $ and a suitable almost complex structure, the corresponding Floer chain complex is generated by the $ (N, \varphi) $-contractible leafwise fixed points. I also outline the construction of a local Floer homology for an arbitrary closed coisotropic submanifold.

Results by Floer and Albers about Lagrangian Floer homology imply lower bounds on the number of leafwise fixed points. This reproduces earlier results of mine.

The first construction also gives rise to a Floer homology for a Boothby-Wang fibration, by applying it to the circle bundle inside the associated complex line bundle. This can be used to show that translated points exist.

Keywords: Floer homology, coisotropic submanifold, leafwise fixed point, leafwise intersection, translated point.
Mathematics Subject Classification: Primary: 53D40.
Citation: Fabian Ziltener. Note on coisotropic Floer homology and leafwise fixed points. Electronic Research Archive, 2021, 29 (4) : 2553-2560. doi: 10.3934/era.2021001
References:
[1]

P. Albers, A note on local floer homology, arXiv: math/0606600. Google Scholar

[2]

P. Albers, A Lagrangian Piunikhin-Salamon-Schwarz morphism and two comparison homomorphisms in Floer homology, Int. Math. Res. Not. IMRN, (2008), Art. ID rnm134, 56 pp. doi: 10.1093/imrn/rnm134.  Google Scholar

[3]

Yu. V. Chekanov, Lagrangian intersections, symplectic energy, and areas of holomorphic curves, Duke Math. J., 95 (1998), 213-226.  doi: 10.1215/S0012-7094-98-09506-0.  Google Scholar

[4]

K. Cieliebak, A. Floer, H. Hofer and K. Wysocki, Applications of symplectic homology, II, Stability of the action spectrum, Math. Z., 223 (1996), 27-45.  doi: 10.1007/BF02621587.  Google Scholar

[5]

A. Floer, Morse theory for Lagrangian intersections, J. Differential Geom., 28 (1988), 513-547.  doi: 10.4310/jdg/1214442477.  Google Scholar

[6]

A. Floer, The unregularized gradient flow of the symplectic action, Comm. Pure Appl. Math., 41 (1988), 775-813.  doi: 10.1002/cpa.3160410603.  Google Scholar

[7]

A. Floer, Symplectic fixed points and holomorphic spheres, Comm. Math. Phys., 120 (1989), 575-611.  doi: 10.1007/BF01260388.  Google Scholar

[8]

H. Geiges and A. I. Stipsicz, Contact structures on product five-manifolds and fibre sums along circles, Math. Ann., 348 (2010), 195-210.  doi: 10.1007/s00208-009-0472-z.  Google Scholar

[9]

V. L. Ginzburg and B. Z. Gürel, Local Floer homology and the action gap, J. Symplectic Geom., 8 (2010), 323-357.  doi: 10.4310/JSG.2010.v8.n3.a4.  Google Scholar

[10]

V. L. Ginzburg and B. Z. Gürel, Fragility and persistence of leafwise intersections, Math. Z., 280 (2015), 989-1004.  doi: 10.1007/s00209-015-1459-y.  Google Scholar

[11]

A. Kapustin and D. Orlov, Remarks on $A$-branes, mirror symmetry, and the Fukaya category, J. Geom. Phys., 48 (2003), 84-99.  doi: 10.1016/S0393-0440(03)00026-3.  Google Scholar

[12]

C.-M. Marle, Sous-variétés de rang constant d'une variété symplectique, Astérisque, 107–108, Soc. Math. France, Paris (1983), 69–86.  Google Scholar

[13]

Y.-G. Oh, Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks. I., Comm. Pure Appl. Math., 46 (1993), 949-993.  doi: 10.1002/cpa.3160460702.  Google Scholar

[14]

Y.-G. Oh, Addendum to: "Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks. I.", Comm. Pure Appl. Math., 48 (1995), 1299-1302.   Google Scholar

[15]

Y.-G. Oh, Floer cohomology, spectral sequences, and the Maslov class of Lagrangian embeddings, Internat. Math. Res. Notices, (1996), 305–346. doi: 10.1155/S1073792896000219.  Google Scholar

[16]

Y.-G. Oh, Localization of Floer homology of engulfed topological Hamiltonian loop, Commun. Inf. Syst., 13 (2013), no. 4, 399–443. Google Scholar

[17] Y.-G. Oh, Symplectic Topology and Floer Homology, Vol. 2, Floer homology and its applications, New Mathematical Monographs, 29, Cambridge University Press, Cambridge, 2015.   Google Scholar
[18]

M. Poźniak, Floer homology, Novikov rings and clean intersections, Northern California Symplectic Geometry Seminar, 119–181, Amer. Math. Soc. Transl. Ser. 2, 196, Adv. Math. Sci., 45, Amer. Math. Soc., Providence, RI, 1999. doi: 10.1090/trans2/196/08.  Google Scholar

[19]

S. Sandon, A Morse estimate for translated points of contactomorphisms of spheres and projective spaces, Geom. Dedicata, 165 (2013), 95-110.  doi: 10.1007/s10711-012-9741-1.  Google Scholar

[20]

F. Ziltener, Coisotropic submanifolds, leaf-wise fixed points, and presymplectic embeddings, J. Symplectic Geom., 8 (2010), 95-118.  doi: 10.4310/JSG.2010.v8.n1.a6.  Google Scholar

[21]

F. Ziltener, A Maslov map for coisotropic submanifolds, leaf-wise fixed points and presymplectic non-embeddings, arXiv: 0911.1460. Google Scholar

[22]

F. Ziltener, Leafwise fixed points for $C^0$-small Hamiltonian flows, Int. Math. Res. Not. IMRN, (2019), 2411–2452. doi: 10.1093/imrn/rnx182.  Google Scholar

show all references

References:
[1]

P. Albers, A note on local floer homology, arXiv: math/0606600. Google Scholar

[2]

P. Albers, A Lagrangian Piunikhin-Salamon-Schwarz morphism and two comparison homomorphisms in Floer homology, Int. Math. Res. Not. IMRN, (2008), Art. ID rnm134, 56 pp. doi: 10.1093/imrn/rnm134.  Google Scholar

[3]

Yu. V. Chekanov, Lagrangian intersections, symplectic energy, and areas of holomorphic curves, Duke Math. J., 95 (1998), 213-226.  doi: 10.1215/S0012-7094-98-09506-0.  Google Scholar

[4]

K. Cieliebak, A. Floer, H. Hofer and K. Wysocki, Applications of symplectic homology, II, Stability of the action spectrum, Math. Z., 223 (1996), 27-45.  doi: 10.1007/BF02621587.  Google Scholar

[5]

A. Floer, Morse theory for Lagrangian intersections, J. Differential Geom., 28 (1988), 513-547.  doi: 10.4310/jdg/1214442477.  Google Scholar

[6]

A. Floer, The unregularized gradient flow of the symplectic action, Comm. Pure Appl. Math., 41 (1988), 775-813.  doi: 10.1002/cpa.3160410603.  Google Scholar

[7]

A. Floer, Symplectic fixed points and holomorphic spheres, Comm. Math. Phys., 120 (1989), 575-611.  doi: 10.1007/BF01260388.  Google Scholar

[8]

H. Geiges and A. I. Stipsicz, Contact structures on product five-manifolds and fibre sums along circles, Math. Ann., 348 (2010), 195-210.  doi: 10.1007/s00208-009-0472-z.  Google Scholar

[9]

V. L. Ginzburg and B. Z. Gürel, Local Floer homology and the action gap, J. Symplectic Geom., 8 (2010), 323-357.  doi: 10.4310/JSG.2010.v8.n3.a4.  Google Scholar

[10]

V. L. Ginzburg and B. Z. Gürel, Fragility and persistence of leafwise intersections, Math. Z., 280 (2015), 989-1004.  doi: 10.1007/s00209-015-1459-y.  Google Scholar

[11]

A. Kapustin and D. Orlov, Remarks on $A$-branes, mirror symmetry, and the Fukaya category, J. Geom. Phys., 48 (2003), 84-99.  doi: 10.1016/S0393-0440(03)00026-3.  Google Scholar

[12]

C.-M. Marle, Sous-variétés de rang constant d'une variété symplectique, Astérisque, 107–108, Soc. Math. France, Paris (1983), 69–86.  Google Scholar

[13]

Y.-G. Oh, Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks. I., Comm. Pure Appl. Math., 46 (1993), 949-993.  doi: 10.1002/cpa.3160460702.  Google Scholar

[14]

Y.-G. Oh, Addendum to: "Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks. I.", Comm. Pure Appl. Math., 48 (1995), 1299-1302.   Google Scholar

[15]

Y.-G. Oh, Floer cohomology, spectral sequences, and the Maslov class of Lagrangian embeddings, Internat. Math. Res. Notices, (1996), 305–346. doi: 10.1155/S1073792896000219.  Google Scholar

[16]

Y.-G. Oh, Localization of Floer homology of engulfed topological Hamiltonian loop, Commun. Inf. Syst., 13 (2013), no. 4, 399–443. Google Scholar

[17] Y.-G. Oh, Symplectic Topology and Floer Homology, Vol. 2, Floer homology and its applications, New Mathematical Monographs, 29, Cambridge University Press, Cambridge, 2015.   Google Scholar
[18]

M. Poźniak, Floer homology, Novikov rings and clean intersections, Northern California Symplectic Geometry Seminar, 119–181, Amer. Math. Soc. Transl. Ser. 2, 196, Adv. Math. Sci., 45, Amer. Math. Soc., Providence, RI, 1999. doi: 10.1090/trans2/196/08.  Google Scholar

[19]

S. Sandon, A Morse estimate for translated points of contactomorphisms of spheres and projective spaces, Geom. Dedicata, 165 (2013), 95-110.  doi: 10.1007/s10711-012-9741-1.  Google Scholar

[20]

F. Ziltener, Coisotropic submanifolds, leaf-wise fixed points, and presymplectic embeddings, J. Symplectic Geom., 8 (2010), 95-118.  doi: 10.4310/JSG.2010.v8.n1.a6.  Google Scholar

[21]

F. Ziltener, A Maslov map for coisotropic submanifolds, leaf-wise fixed points and presymplectic non-embeddings, arXiv: 0911.1460. Google Scholar

[22]

F. Ziltener, Leafwise fixed points for $C^0$-small Hamiltonian flows, Int. Math. Res. Not. IMRN, (2019), 2411–2452. doi: 10.1093/imrn/rnx182.  Google Scholar

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